#6455463

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1497 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,29)(20,26)(21,27)(22,25)(23,28)(24,33)(30,38)(31,45)(32,43)(34,48)(35,40)(36,42)(37,44)(39,47)(41,46);(2,5)(6,8)(17,19,22,33)(18,24,25,29)(20,34,35,46)(21,32,23,30)(26,41,40,48)(27,38,28,43)(31,37,47,36)(39,44,45,42);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,25)(19,30)(20,35)(21,23)(24,38)(29,43)(31,46)(32,33)(34,47)(39,41)(42,44)(45,48);(17,20,22,35)(18,26,25,40)(19,31,33,47)(21,36,23,37)(24,39,29,45)(27,42,28,44)(30,34,32,46)(38,48,43,41);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,25,28)(19,32,33,30)(20,36,35,37)(24,38,29,43)(26,42,40,44)(31,46,47,34)(39,48,45,41)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362659);